Constructing Regular Lovelock Black Holes with degenerate vacuum and $Λ< 0$, using the gravitational tension. Shadow analysis

TL;DR

This work extends Lovelock gravity to LnFDGS, an $n$-fold degenerate AdS ground state, to realize regular black holes with a negative cosmological constant without curvature singularities. By redefining gravitational tension through a Poincaré AdS-like curvature and introducing an AdS-compatible Kretschmann scalar $K'$, the authors construct a finite-energy density that yields a non-singular core and asymptotically AdS behavior, with both analytic and numerical treatment of horizons, photon spheres, shadows, and thermodynamics. A key result is that, for masses above the extremal value $M_{\min}$, the regular LnFDGS AdS black hole becomes indistinguishable from its vacuum counterpart outside the horizon, while at small scales quantum-like effects and the matter source remove singularities and produce a remnant with $T\to0$ at $M_{\min}$. The paper develops a numerical methodology to relate event horizons, photon spheres, and shadows in this setup, highlighting the rich interplay between geometry, thermodynamics, and observational signatures in higher-curvature gravity with AdS asymptotics.

Abstract

In \cite{Estrada:2024uuu}, a link between gravitational tension (GT) and energy density via the Kretschmann scalar (KS) was proposed to construct regular black holes (RBHs) in Pure Lovelock (PL) gravity. However, including a negative cosmological constant in PL gravity leads to a curvature singularity \cite{Cai:2006pq}. Here, we choose the coupling constants such that the Lovelock equations admit an $n$-fold degenerate AdS vacuum (LnFDGS), allowing us to construct an RBH with $Λ< 0$, where the energy density is analogous to the previously mentioned model. To achieve this, we propose alternative definitions for both the KS and GT. We find that, for mass parameter values greater than the extremal value $M_{\text{min}}$, our RBH solution becomes indistinguishable from the AdS vacuum black hole from inside the event horizon out to infinity. At small scales, quantum effects modify the geometry and thermodynamics, removing the singularity. Furthermore, due to the lack of analytical relationships between the event horizon, photon sphere, and shadow in LnFDGS, we propose a numerical method to represent these quantities.

Figures (6)

• Figure 1: In horizontal axis $\bar{r}$. In vertical axis: $M$ of our RBH is shown in blue and $M$ in the vacuum case Aros:2000ij in dashed red. We have used $d = 6,n = 2,l = 10,a = 1$. We can check that this behavior is generic by using other parameter values. The decreasing part of the blue curve corresponds to the inner horizon of the RBH solution, while the increasing part represents the event horizon. The minimum point marks the extremal radius. The red curve represents the vacuum AdS solution. For a value slightly greater than the extremal radius, both solutions become indistinguishable for radial values equal to or greater than the event horizon.
• Figure 2: On the horizontal axis, $r$. On the vertical axis, the function $f(r)$ for $M = M_{\text{ext}} \approx 1.01$, $M = 1.2 > M_{\text{ext}}$, and $M = 2 > M_{\text{ext}}$ in the first, second, and third panels, respectively. We have used $d = 6$, $n = 2$, $l = 10$, $a = 1$. We can check that this behavior is generic for other parameter values. We observe that for mass parameter values greater than the extremal mass, $M > M_{\text{ext}}$, both solutions become indistinguishable from within the event horizon.
• Figure 3: Scheme: We observe that photons launched with an impact parameter smaller than the shadow radius fall into the black hole. Photons with an impact parameter equal to the shadow radius fall into an unstable circular orbit. Photons launched with an impact parameter greater than the shadow radius escape toward the boundary of spacetime.
• Figure 9: This figure displays the values of the photon sphere radius corresponding to different values of the event horizon radius. The values on the horizontal axis represent the event horizon radius $r_+$. The values on the vertical axis represent the sphere photon radius $r_{sp}$. In left panel we have used $n=2, a=4,l=100$, $d=6$(red), $d=7$(blue), $d=8$(black). In right panel we have used $n=3, a=4,l=100$, $d=8$(red), $d=9$(blue), $d=10$(black)
• Figure 10: This figure displays the values of the shadow radius corresponding to different values of the photon sphere radius. The values on the horizontal axis represent the sphere photon radius $r_{sp}$. The values on the vertical axis represent the shadow radius $r_{sh}$. In left panel we have used $n=2, a=4,l=100$, $d=6$(red), $d=7$(blue), $d=8$(black). In right panel we have used $n=3, a=4,l=100$, $d=8$(red), $d=9$(blue), $d=10$(black)